Backaction-evading measurement of entanglement in optomechanics

We propose here a fully backaction-evading scheme for the measurement of the entanglement between two nanomechanical resonators. The system, which consists of two mechanical oscillators, coupled to a single mode of an electromagnetic resonant cavity through a radiation-pressure interaction term, is driven by two pump tones and four detection tones. As previously discussed in the literature, the former induce entanglement between the two mechanical oscillators, while we show here that a specific choice of phase and amplitude of the detection tones allows for direct pairwise reconstruction of the collective quadrature fluctuations of the mechanical oscillators belonging to quantum-mechanics-free subspaces, thereby providing direct evidence of the entanglement properties of the two mechanical resonators.

Figure 1

(a) Sketch of the proposed entanglement detection scheme. The pump tones are depicted as blue (solid dark) arrows, while the detection tones are depicted as wiggly dashed lines. In addition, we have indicated the three sources of noise $\left(a_{\mathrm{I}}^{\mathrm{in}},a_{\mathrm{E}}^{\mathrm{in}},b^{\mathrm{in}}\right)$ as short wiggly arrows. (b) Pictorial representation of the cavity spectrum corresponding to the choice ${\omega }_{+}={\omega }_1,{\omega }_{-}={\omega }_2,{\omega }_{\mathrm{\Sigma }}=\left({\omega }_1+{\omega }_2\right)/2,\delta =\left({\omega }_1-{\omega }_2\right)/2$ (frequency in the rotating frame; see text). The cavity mode is driven with two pumps (blue) and generates two sidebands at $\pm \delta$. In addition to the strong driving tone, we consider four probing tones (gray) which generate sidebands at $\pm 2\delta$ and 0. In our analysis we focus on the peak generated at 0, which, as we will show, contains all the required information to ascertain the violation of the Duan bound.

Figure 2

(a) Noise spectrum of the symmetrical mechanical quadrature $S_{\omega }^{\mathrm{\Sigma },\theta }$ as a function of frequency $\omega$ for $\theta =0$ (red dotted curve) and $\theta =\pi /2$ (blue solid curve). (b) Spectrum of the antisymmetrical mechanical quadrature $S_{\omega }^{\mathrm{\Delta },\theta }$ for $\theta =0$ (red dotted curve) and $\theta =\pi /2$ (blue solid curve). Parameters are $\gamma ={10}^{-5},\delta =0.1,G_{-}=4.8\times {10}^{-2},G_{+}=4.0\times {10}^{-2}$, and $n_1=n_2=10,n_{\mathrm{c}}=0$. All frequencies are in units of $\kappa ,\hslash =1$ throughout the paper.

Figure 3

(a) Output spectrum ${\left.S_{\omega }^{\mathrm{out}}\right|}_{\overline{\theta }=0}$ as a function of frequency for ${\varphi }_{\mathrm{p}}=0,{\varphi }_{\mathrm{q}}=0$ (red dotted curve) and ${\varphi }_{\mathrm{p}}=0,{\varphi }_{\mathrm{q}}=0$ (blue solid curve). (b) Output spectrum ${\left.S_{\omega }^{\mathrm{out}}\right|}_{\overline{\theta }=\pi /2}$ as a function of frequency for ${\varphi }_{\mathrm{p}}=0,{\varphi }_{\mathrm{q}}=\pi$ (red dotted curve) and ${\varphi }_{\mathrm{p}}=\pi /2,{\varphi }_{\mathrm{q}}=-\pi /2$ (blue solid curve). ${\kappa }_{\mathrm{E}}=0.9$; all other parameters as in Fig. 2. The quantity appearing in the Duan inequality given by Eq. (1) can be inferred from the area under the red dotted curve in (a) and the blue solid curve in (b).

Figure 4

Plot of the Duan quantity in Eq. (1) as a function of ratio $G_{+}/G_{-}$. The dashed line indicates the threshold below which the Duan inequality is violated. All parameters except $G_{+}$ as in Figs. 2 and 3. As discussed in the text, the value of the Duan quantity can be extracted from the output spectrum as the sum of the integral under the red dotted curve in Fig. 3 and the blue solid curve in Fig. 3. See also Table 1. The values of ${\varphi }_{\mathrm{p}}$ and ${\varphi }_{\mathrm{q}}$ chosen here imply in all cases that ${\varphi }_1-{\varphi }_2=0,\pi$, since ${\varphi }_{1,2}=arctan[\frac{u+v}{u-v}tan\left({\varphi }_{\mathrm{p},\mathrm{q}}\right)]$.